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# Sharedchao

**Example Calculations**

***.sharedChao**

Example calculations below will be performed using data from the Eckburg 70.stool_compare files with an OTU definition of 0.03.

Estimating the richness of shared OTUs between two communities. A Non-parametric richness estimator of the number of shared OTUs between two communities has been developed that is analogous to the Chao1 (6) single community richness estimator. The <math>S_{A,B Chao} \mbox{(11)</math> estimator is calculated as:

<math>S_{A,B Chao} = S_{12 \left ( Obs \right )} + f_{11} \frac {f_{1+}f_{+1}}{4f_{2+}f_{+2}} + \frac{f_{1+}^{2}}{2f_{2+}} + \frac{f_{+1}^{2}}{2f_{+2}}</math>,

where,

<math>f_{11}</math> = number of shared OTUs with one observed individual in A and B

<math>f_{1+}, f_{2+}</math> = number of shared OTUs with one or two individuals observed in A

<math>f_{+1}, f_{+2}</math> = number of shared OTUs with one or two individuals observed in B

<math>f_{\left(rare \right)1+}</math> = number of OTUs with one individual found in A and less than or equal to 10 in B.

<math>f_{\left(rare \right)+1}</math> = number of OTUs with one individual found in B and less than or equal to 10 in A.

<math>n_{rare}</math> = number of sequences from A that contain less than 10 sequences.

<math>m_{rare}</math> = number of sequences from B that contain less than 10 sequences.

<math>S_{12\left(rare\right)}</math> = number of shared OTUs where both of the communities are represented by less than or equal to 10 sequences.

<math>S_{12\left(abund\right)}</math> = number of shared OTUs where at least one of the communities is represented by more than 10 sequences.

<math>S_{12\left(obs\right)}</math> = number of shared OTUs in A and B.

Calculation of the <math>S_{A,B Chao}</math> requires the number of OTUs where only one sequence was observed from each library, <math>f_{11}</math>. For our example case, <math>f_{11}</math> is 2. Plugging the f-values and <math>S_{12\left( obs \right )}\mbox{into } S_{A,B Chao}</math> yields a value of 76.7, which matches the value in <math>S_{A,B Chao}</math> of the table above.